On the Strong Approximation for a Simple Reentrant Line in Light Traffic Under First-buffer First-served Service Discipline

For a 2-station and 3-class reentrant line under first-buffer first-served(FBFS)service discipline in light traffic,we firstly construct the strong approximations for performance measures including the queue length,workload,busy time and idle time processes.Based on the obtained strong approximations,we use a strong approximation method to find all the law of the iterated logarithms(LILs)for the above four performance measures,which are expressed as some functions of system parameters:means and variances of interarrival and service times,and characterize the fluctuations around their fluid approximations.

Variability Analysis for a Two-station Queueing Network in Heavy Traffic with Arrival Processes Driven by Queues

The law of the iterated logarithm(LIL)for the performance measures of a two-station queueing network with arrivals modulated by independent queues is developed by a strong approximation method.For convenience,two arrival processes modulated by queues comprise the external system,all others are belong to the internal system.It is well known that the exogenous arrival has a great influence on the asymptotic variability of performance measures in queues.For the considered queueing network in heavy traffic,we get all the LILs for the queue length,workload,busy time,idle time and departure processes,and present them by some simple functions of the primitive data.The LILs tell us some interesting insights,such as,the LILs of busy and idle times are zero and they reflect a small variability around their fluid approximations,the LIL of departure has nothing to do with the arrival process,both of the two phenomena well explain the service station’s situation of being busy all the time.The external system shows us a distinguishing effect on the performance measures:an underloaded(overloaded,critically loaded)external system affects the internal system through its arrival(departure,arrival and departure together).In addition,we also get the strong approximation of the network as an auxiliary result.

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery,blood vessel or circulatory system with experimental support

Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in,for example,variable-radius arteries.With respect to the nonlinear waves in an artery full of blood with certain aneurysm,pulses in a blood vessel,or features in a circulatory system,this paper symbolically computes out an auto-B?cklund transformation via a noncharacteristic movable singular manifold,certain families of the solitonic solutions,as well as a family of the similarity reductions for a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation.Aiming,e.g.,at the dynamical radial displacement superimposed on the original static deformation from an arterial wall,our results rely on the axial stretch of the injured artery,blood as an incompressible Newtonian fluid,radius variation along the axial direction or aneurysmal geometry,viscosity of the fluid,thickness of the artery,mass density of the membrane material,mass density of the fluid,strain energy density of the artery,shear modulus,stretch ratio,etc.We also highlight that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.

Fluid Approximation and Its Convergence Rate for GI/G/1 Queue with Vacations

A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed （i.i.d.） random variables, that is max{ηi, 1 ≤ i ≤ n}. The approximating technique is used to obtain the fluid approximation for the queue length, workload and busy time processes. Furthermore, under uniform topology, if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate, we prove by the approximating technique that the scaled processes characterizing the queue converge to the corresponding fluid limits with the exponential rate only for large N. Here the scaled processes include the queue length process, workload process and busy time process.

Bilinear forms through the binary Bell polynomials, N solitons and Backlund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.

Asymptotic Variability Analysis for Multi-Server Generalized Jackson Network in Overloaded

The asymptotic variability analysis is studied for multi-server generalized Jackson network. It is characterized by law of the iterated logarithm （LIL）, which quantifies the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. In the overloaded （OL） case, the asymptotic variability is studied for five performance measures： queue length, workload, busy time, idle time and number of departures. The proof is based on strong approximations, which approximate discrete performance processes with （reflected） Brownian motions. We conduct numerical examples to provide insights on these LIL results.